Thursday, 20 November 2014

The Eyam Plague of 1666: A System Dynamics Model

At present the news is full of the rather frightening Ebola
outbreaks in West Africa. Other infectious diseases also get into the
news often, for example AIDS/HIV and flu. What is less well know is
that such infectious diseases spread according to fairly precise mathematical
rules. This follows from the person-to-person contact involved in the spread of
the disease.

It is this process that the limited enthusiasm hypothesis of
the church growth models is based on. In that case the “disease” is faith, and
it is spread by word of mouth contact. It is not only churches that grow this
way, the same epidemiological mechanism has been used to model the spread of
languages, scientific ideas, riot behaviour, bulimia, cigarette smoking and
even Facebook [1].

To illustrate how this principle works I want to use a
standard case study in mathematical epidemiology: the spread of the plague in
the Derbyshire village of Eyam in 1666.

Eyam Plague

The years 1665-1666 saw the great plague hit England,
notably London, which lost about 15% of its population [2]. This was the last
epidemic of bubonic plague in the UK, a disease that had been an ongoing
problem since the days of the Black Death in the 14th century. The
primary mechanism of spread of the disease is through the bite of an infected
black rat flea. However once established the disease can spread person-to-person,
which gave rise to the popular rhyme “Ring a ring of roses, a pocket full of
posies, atishoo, atishoo, we all fall down” [3].

Although largely confined to London, an outbreak occurred in
the Derbyshire village of Eyam due to a person acquiring the disease from a
piece of infected cloth sent from London in 1665. Once the Eyam outbreak took
hold in 1666 the local clergyman took the precaution of isolating the village,
as best he could, to prevent the spread of the disease. This action of his made
it an ideal case study to mathematically model the spread of a disease, as
migration could be ruled out as a major mechanism in its spread.

Such a mathematical model of the plague was carried out by
GF Raggett [4] using methods based on differential equations. Raggett explained
why the spread of the disease in Eyam must have been largely person-to-person
rather than rat fleas, as over the period of a year infected rats would have
left the area and infected the wider area. No such cases occurred. Using
mathematics Raggett then showed how the model predicted the number of deaths due to
the epidemic, and demonstrated some important results [5].

What follows is a system dynamics version of Raggett’s work
to help explain how a disease spreads without using mathematics. The model is
often called the SIR model, after the symbols in the equations, the epidemic
model, or the Kermack McKendrick model, after the first people who published it
[6].

System Dynamics Model

The model assumes the population of people are split into 3
categories of people: the Susceptibles,
who could potentially catch the disease; the Infected, who are carrying the disease and could infect others; and
the Removed who have had the disease
and cannot catch it again, either because they are cured and immune, or have
died. The letters SIR stand for these three categories.

In system dynamics this model can be expressed as stocks and
flows:

Figure 1

The removed category has been renamed Deceased as most cases of bubonic plague ended in death.

There are two processes (called flows) involved: catch the disease, which moves susceptibles
into infected; and deaths, which
moves infected into deceased.

The catch disease
process is subject to two social forces: R1
and B1. R1 causes the increase in the number of infected to accelerate as
more infected gives more new cases each day, thus more infected. This is called
reinforcing feedback and is the first phase of growth in the infected, (figure
2).

Figure 2

In addition the force B1
slows that growth as the pool of susceptibles is depleted, making it harder for
infected people to make new cases. This slowing force is balancing feedback and
opposes the force R1. B1 eventually dominates over R1, the second phase of growth (figure
2) [7].

Eventually the number
who catch the disease drops below the deaths
and B1 now causes the infected to
decline faster and faster, the first phase of decline (figure 2).

The deaths process
is subject the social force B2 as the
more infected there are the more die, thus depleting their numbers. This force
only dominates at the end causing the decrease in infected to slow down, the
second phase of decline, (figure 2).

Raggett [4] showed from the recorded deaths that when the
main period of the plague epidemic started, June 19th 1666, there
were 7 people infected. The population was known to be 261 at that time. By the
end of the epidemic, in the middle of October that year, only 83 people had
survived.

From these figures, and knowing the infectious period of the
plague is about 11 days, it is possible to simulate the system dynamics model,
and compare it with the data for cumulative deaths (green curve, figure 3) [8].
Comparing Deceased with recorded
deaths shows a good fit. It is remarkable that something that involves people,
and random behaviour, gives such predictable results. This predictability is
what allows modern day epidemics to be so successfully tackled, and the
consequences of not tacking action computed [9].

Figure 3

Note the following:

1. The epidemic burns out before everyone gets the
disease. There are susceptibles remaining at the end of the epidemic (black
curve figure 3).

2. At the peak of the epidemic, where the number of
daily cases is at a maximum (about 45 days, figure 3), even though the daily
death rate is slowing down the epidemic is not at an end and a significant
number of deaths are still to come, (green curve figure 3).

3. At any one time the number of infected people is
quite small compared with the population (blue curve, figure 2 and figure 3). It
is their cumulative number over time that is large. It does not take many
infected people at a given time to keep an epidemic going [10].

Reproductive Ratio

The strength of an epidemic is measured by the reproductive
ratio, called R0. At its simplest it
is the number of people one infected person could potentially infect during
their infectious period, if the whole population were susceptible [11]. The
number has to be bigger than one for an epidemic to happen. The larger this
number then the bigger the epidemic becomes. Different diseases have different
reproductive ratios [12].

Using the numbers above, the reproductive ratio comes out at
about R0 = 1.6 [13]. This is much
less than highly infectious diseases such as Measles (range 12-18) and Smallpox
(5-7) [14]. Nevertheless 1.6 was still large enough for well over half the
population of Eyam to get the disease. A value of R0 of 1.6 is similar to Ebola (1.5-2.5). However because Bubonic
Plague is spread through fleas, and through the air, it is harder to take
action to reduce R0 compared with
Ebola, which is only spread through contact with bodily fluids.

Conclusion

What turned out to be an ideal case study to test a
mathematical model for the spread of a disease turned out to be a tragedy for
the people of Eyam. The majority of the population died, including the wife of
the brave clergyman who isolated the village and performed all the burials [15].
However his action saved many more lives of people in the region, and the
lessons learned, which mathematicians can now explore, gives confidence to
models that have given strategies to combat epidemics and save millions of
people. That studies of this sort can help understand social diffusion
processes such as religion is a bonus.

[7] The structure of the feedback loops, or forces, on those
who catch the disease can be broken down into a number of parts where it is
assumed the populations are proportionally mixed. The connection between
population density and the likelihood of contact requires further assumptions.

[10] All these epidemiological principles are replicated in
church growth, and other forms of social diffusion. Not all people in a
population are converted. Substantial church growth still comes after the peak
in the growth is over. At any one time there are very few infected, called
enthusiasts, spreading the faith.

[11]The
reproductive ratio (or reproductive number) is called the reproduction potential in church growth and
measures how many people one enthusiast can potentially convert and make
an enthusiast. Not all converts become enthusiasts.

[12] For most diseases the reproductive ratio is given as a
range as its value can depend on population density and particular population
behaviours. It is believed that Ebola in West Africa in 2014 started with a
much higher than normal R0 due the
particular burial practices used, allowing dead bodies to transmit the disease,
thus extending the infectious period.

[13] A simple formula can be used to compute the
reproductive ratio in terms of the population number, and initial and final
number of susceptibles alone. This calculation was done in the software Mathcad:

[15] There is a museum in Eyam where the visitor can learn
about the outbreak. http://www.eyam-museum.org.uk/
Note that there had been cases and deaths in 1665 and early 1666 before the
period of study used by Raggett starting June 19 1666. Thus the total deaths,
and the original village size, are larger than used in Raggett’s study.